Assumption REF  is a standard assumption in SQP analysis [1]}, [2]}.
The compactness of \(\mathcal {X}\times \Lambda \)  ensures that \(\mathcal {L}_{\mu , \nu }\)  is lower bounded, and there exists a constant \(\Upsilon _u\ge 1\)  such that
\(\Vert \nabla ^2\mathcal {L}({x}, {\lambda })\Vert  \vee \Vert \nabla \mathcal {L}({x}, {\lambda })\Vert \vee \Vert \nabla f({x})\Vert  \le \Upsilon _u, \quad \text{ for any } ({x}, {\lambda })\in \mathcal {X}\times \Lambda .\) 
